### Abstract

We present an evolutionary self-governing model based on the numerical atomic rule Z(a,b)=ab/gcd^{(a,b)2}, for a, b positive integers. Starting with a sequence of numbers, the initial generation _{Gin}, a new sequence is obtained by applying the Z-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix _{T}_{Gin} is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If _{Gin} is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called Z-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that _{T}N∗ is close to the analogue free of Z-solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix _{T}N∗. We solve one of them and present arguments for a precise conjecture on another.

Original language | English (US) |
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Pages (from-to) | 136-147 |

Number of pages | 12 |

Journal | Chaos, solitons and fractals |

Volume | 91 |

DOIs | |

State | Published - Oct 1 2016 |

### Keywords

- Absolute differences
- Cellular automata
- Discrete solitons
- Dynamic lattice systems
- Gros sequence
- Growth model
- Recurrent sequences
- Ruler sequence
- Sandpiles
- Self-similar processes
- Z-solitons

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)
- Physics and Astronomy(all)
- Applied Mathematics

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## Cite this

*Chaos, solitons and fractals*,

*91*, 136-147. https://doi.org/10.1016/j.chaos.2016.05.016